182
W.H.F. Smith
This chapter reviews those aspects of potential theory, gravity and geodesy that
are relevant for the present accuracy and error in the marine geoid. These include
the relation between gravity anomaly potential and geoid height, upward continuation, and the use of ship gravimetry to verify marine gravity field models. The
power spectrum of the gravity field is shown in comparison with Earth topography,
to establish that the marine geoid has considerable signal due to sea floor topography. This signal cannot be sensed at orbital altitude by space gravity missions,
due to upward continuation. The geoid slope at short scales can be determined from
altimetry, and I present an assessment of this using marine gravity surveys, as well
as a summary of other tests and results presented elsewhere.
11.2 The Geoid in Classical Potential Theory
In classical geodesy, the geoid height is related to the gravity potential as follows.
Here, “potential” is used to mean potential energy per unit mass. Let V be the
Newtonian gravitational attraction potential of the Earth. Let W = V + be the
gravity potential, with Φ the potential of the Earth’s rotation, assumed uniform.
Let a reference ellipsoidal gravity field model U = E + be defined, with E
the attraction potential and Φ again accounting for the rotation; E is chosen so that
when U = U 0 , a constant, the shape of the U = U 0 surface is an oblate ellipsoid of
revolution corresponding closely to the Earth’s actual geoid. The reference gravity
acceleration is
γ = ∇U. Thus on the ellipsoid the normal to the ellipsoid would be
the direction of the vertical in the absence of gravity anomalies.
Let φ, λ, z indicate the latitude, longitude and height above the reference ellipsoid, and make a first-order Taylor series expansion of the gravity potential at a
height z in terms of the value on the ellipsoid:
W (φ,λ,z) = W (φ,λ,0) + z ˆ
n · ∇W (φ,λ,0) + ...
(11.1)
Setting W (φ, λ, z) = U 0 in this equation gives an implicit equation for the geoid
height anomaly, z = N (φ, λ).
The potential of the actual field departs from the ellipsoidal model field by an
amount, T = W − U = V − E, known as the disturbing potential in the literature.
T is the potential of the anomalies in the gravity field. Since T is smaller than 10 –3
times W, in solving (11.1) for the geoid height we may make the approximations
γ ≈ ∇W and ˆ
n · ∇W ≈ −γ 0 , with γ 0 the magnitude of reference (standard) gravity
on the ellipsoid. Then the expression for the geoid height reduces to
N (φ, λ) ≈
T (φ, λ, 0)
γ 0 (φ)
(11.2)
which is known as Bruns’ formula (Bruns, 1828).
Two approximations are made in Bruns’ formula. First, the Taylor series (11.1)
is truncated after the first-order term. Second, the actual gravity is replaced by the
W.H.F. Smith
This chapter reviews those aspects of potential theory, gravity and geodesy that
are relevant for the present accuracy and error in the marine geoid. These include
the relation between gravity anomaly potential and geoid height, upward continuation, and the use of ship gravimetry to verify marine gravity field models. The
power spectrum of the gravity field is shown in comparison with Earth topography,
to establish that the marine geoid has considerable signal due to sea floor topography. This signal cannot be sensed at orbital altitude by space gravity missions,
due to upward continuation. The geoid slope at short scales can be determined from
altimetry, and I present an assessment of this using marine gravity surveys, as well
as a summary of other tests and results presented elsewhere.
11.2 The Geoid in Classical Potential Theory
In classical geodesy, the geoid height is related to the gravity potential as follows.
Here, “potential” is used to mean potential energy per unit mass. Let V be the
Newtonian gravitational attraction potential of the Earth. Let W = V + be the
gravity potential, with Φ the potential of the Earth’s rotation, assumed uniform.
Let a reference ellipsoidal gravity field model U = E + be defined, with E
the attraction potential and Φ again accounting for the rotation; E is chosen so that
when U = U 0 , a constant, the shape of the U = U 0 surface is an oblate ellipsoid of
revolution corresponding closely to the Earth’s actual geoid. The reference gravity
acceleration is
γ = ∇U. Thus on the ellipsoid the normal to the ellipsoid would be
the direction of the vertical in the absence of gravity anomalies.
Let φ, λ, z indicate the latitude, longitude and height above the reference ellipsoid, and make a first-order Taylor series expansion of the gravity potential at a
height z in terms of the value on the ellipsoid:
W (φ,λ,z) = W (φ,λ,0) + z ˆ
n · ∇W (φ,λ,0) + ...
(11.1)
Setting W (φ, λ, z) = U 0 in this equation gives an implicit equation for the geoid
height anomaly, z = N (φ, λ).
The potential of the actual field departs from the ellipsoidal model field by an
amount, T = W − U = V − E, known as the disturbing potential in the literature.
T is the potential of the anomalies in the gravity field. Since T is smaller than 10 –3
times W, in solving (11.1) for the geoid height we may make the approximations
γ ≈ ∇W and ˆ
n · ∇W ≈ −γ 0 , with γ 0 the magnitude of reference (standard) gravity
on the ellipsoid. Then the expression for the geoid height reduces to
N (φ, λ) ≈
T (φ, λ, 0)
γ 0 (φ)
(11.2)
which is known as Bruns’ formula (Bruns, 1828).
Two approximations are made in Bruns’ formula. First, the Taylor series (11.1)
is truncated after the first-order term. Second, the actual gravity is replaced by the